How to Choose a Better Box: Complex Absorbing Potential Optimization for Anionic Resonances

Electronic-structure calculations using complex absorbing potentials (CAPs) to stabilize temporary anion states are very sensitive to the CAP configuration, including the coupling strength (<i>η</i>) and boundary geometry. We present high-resolution surveys of several two-dimensional parameter spaces (CAP spaces for short) for the CO^- (²Π) and N₂^- (²Π_<i>g</i>) resonances and propose an efficient optimization strategy for box-CAPs that relies on Gyamfi and Jagau's <i>ξ</i> error function [<i>J. Chem. Theory Comput.</i> 2024, 20, 1096-1107]. In all CAP spaces probed, only narrow parameter ranges yield the highest-quality results (<i>ξ</i> ∼ 10^-4-10^-5) that minimize wave function reflections and perturbations. Such optimal conditions are unlikely to be found by ad hoc methods, necessitating a systematic optimization protocol. Ours begins with (<i>η</i>, <i>r</i>^o)<i>_L</i> optimization trajectories, where <i>r</i>^o is a geometric variable controlling the CAP boundary and <i>L</i> is a distinct parameter such as the box elongation. Unlike extensively studied <i>η</i>- and <i>r</i>^o-trajectories, the (<i>η</i>, <i>r</i>^o)<i>_L</i> trajectories are optimum-seeking paths in two-dimensional CAP spaces searching for <i>ξ</i> minima. After optimizing CAPs across multiple (<i>η</i>, <i>r</i>^o)<i>_L</i> spaces with varying <i>L</i>, CAP strength minimization along the <i>L</i>-trajectory defines the overall optimal configuration in the higher-dimensional (<i>η</i>, <i>r</i>^o, <i>L</i>) space. This hierarchical optimization, min <i>ξ</i> ≻ min <i>η</i> (where "≻" denotes lexicographic precedence), produces a low-error description of the resonance (min <i>ξ</i>) stabilized by the weakest (but sufficient) CAP possible, i.e., min <i>η</i> subject to the min <i>ξ</i> constraint.